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How to find the maximum value of an exponential function?

User Sugre
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Final answer:

To find the maximum value of an exponential function, you can follow these steps: rewrite the function, find the derivative, set it equal to zero, solve for x, and plug it back in to find the maximum value. An example is provided.

Step-by-step explanation:

To find the maximum value of an exponential function, you can follow these steps:

  1. First, rewrite the exponential function in the form f(x) = ae^mx, where a is a constant and m is the exponent.
  2. Next, take the derivative of the function with respect to x to find the critical points.
  3. Set the derivative equal to zero and solve for x to find the critical point(s).
  4. Plug the critical point(s) back into the original function to find the corresponding y-coordinate(s).
  5. The maximum value of the exponential function will be the highest y-coordinate among the critical point(s) you found.

For example, let's say we have the function f(x) = 2e^(-0.5x). We can find the maximum value by taking the derivative, setting it equal to zero, solving for x, and plugging it back into the original function.

Derivative: f'(x) = -0.5e^(-0.5x)

Critical point: -0.5e^(-0.5x) = 0

x = 0

Plugging x = 0 back into the original function: f(0) = 2e^0 = 2

So the maximum value of the function is 2.

User ILLIA DEREVIANKO
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